Volumetric locking-free Mixed Virtual Element Methods for Contact Problems

TL;DR

This paper introduces volumetric locking-free mixed virtual element methods for 2D contact problems in elasticity, providing robust schemes that handle nearly incompressible materials and complex meshes with small edges, with confirmed theoretical convergence.

Contribution

The paper develops and analyzes new mixed virtual element schemes that prevent volumetric locking and are effective on meshes with small edges, advancing numerical methods for elasticity contact problems.

Findings

Schemes are robust against volumetric parameter $$

Methods perform well with small mesh edges

Achieve expected convergence rates

Abstract

We consider the approximation of the 2D frictionless contact problem in elasticity using the Virtual Element Methods (VEMs). To overcome the volumetric locking phenomenon in the nearly incompressible case, we adopt a mixed displacement/pressure ($u/p$) variational formulation, where pressure is introduced as an independent unknown. We present the VEM discretization and develop a general error analysis, keeping explicit track of the constants involved in the error estimates, thus allowing to consider meshes with "small edges". As examples, we consider two possible VEM schemes: a first-order scheme and a second-order scheme. The numerical results confirm the theoretical predictions, specifically both schemes show: 1) robustness with respect to the volumetric parameter $\lambda$, thus preventing the occurrence of the volumetric locking phenomenon; 2) good behavior even in the presence of "small edges"; 3) achievement of the expected theoretical convergence rates.